The present invention relates generally to techniques for reducing the size of time domain data with minimal distortion.
A uniformly sampled data set of a continuous time based function ƒ(t) is defined as:
  {            f      ⁡              (                  x          i                )              |                                                      x              i                        ∈                          domain              ⁢                                                          ⁢              of              ⁢                                                          ⁢              f                                                                                      and              ⁢                                                          ⁢                                                                                    x                                          i                      +                      1                                                        -                                      x                    i                                                                                        =            κ                                                                          and              ⁢                                                          ⁢                              1                κ                                      ≥                          2              ⁢                                                          *                                                          ⁢              Maximum              ⁢                                                          ⁢              Frequency              ⁢                                                          ⁢              of              ⁢                                                          ⁢              Interest                                            }
The uniformly sampled data may also be called a time sampled representation of ƒ(t). Nyquist proved that a time based function can be reconstructed perfectly from a time series data set if the sampling is greater than twice the maximum frequency. However, this method can require very large amounts of data storage.
For example, a speaker's energy spectrum is usually limited to less than 20 KHz. Thus, a uniformly sampled data set for speech would require a sampling rate at least 40 K samples per second. Using 16 bits to account for dynamic range means that every second of music requires 80 KB. Medical ultrasound, which uses frequencies as high as 20 MHz, needs 40 million samples per second for perfect reconstruction. Using the same dynamic range as music means that a medical ultrasound would generate 80 MB/s of uniform time sampled data. Imagery data can require both larger dynamic range and higher sampling rates.
However, if one has a uniform time sampled data set and does not need perfect reconstruction of the time-based function, the size of the representative data set can be reduced?
Techniques to that reduce the size of the uniform time sampled data set and allow perfect reconstruction of the time-based function are called lossless compression techniques. Techniques that reduced the size of the uniform time sampled data set but do not provided perfect reconstruction of the time-based function are called compaction or lossy compression techniques.
Thus, the problem is how to reduce the size of the time data set using compaction while having minimal differences between the original time-based function and the reconstructed time-base function. It would be desirable to have a technique or algorithm that solves this problem.
Several algorithms exist that allow lossless compression of a uniformly time sampled data set, such as zip, and bzip2. In addition, techniques exist for compacting uniform time sampled data sets, such as eliminating every other sample point (decimation). Decimation effectively reduces the frequency range of the reconstructed time-based function; and may cause distortion due to frequency aliasing. For example, if the maximum frequency range for music is reduced from 20 KHz to 10 KHz by decimating by 2, the quality of the music would be unacceptable because all energy above 5-Hz is aliased.
Thus, the problem transforms into how to reduce the size of the uniformly time sampled data set while maintaining the full frequency range and minimizing the distortion (differences) between the original time-based function and the reconstructed time-based function.
Existing compaction techniques require prior knowledge of the internal structure of the time-based function. For example, techniques exist to eliminate non-clarinet sounds in a uniformly time sampled music data set. The reconstructed time-based function only contains the sounds of the clarinet. However, this is a problem if someone is interested in a trumpet part.
Since there is no prior knowledge of the internal structure of the time-base function, another type of technique is required. Since it is desired to minimize the distortion between the original time-based function and the reconstructed time-based function, the present inventors investigated available techniques.
The most promising techniques involved the use of dual spaces. If one looks to functional analysis, there exists a set of function space transforms that convert one functional space to a new functional space. In particular, the functional space transforms of interest are Laplace transforms. Fourier transforms are one example of Laplace transforms.
Techniques involving filtering of a continuous time-based function in a dual space provide a needed internal structure based on the transformed time-based function. This structure requires no knowledge of internal structure of the time-based function. The frequency domain signal may be viewed as a filtering operation uses overlapping Quadrature Mirror Filtering (QMF), for example, which allow perfect reconstruction when converting back to the time domain.
These techniques are applicable to a uniformed time sampled data set derived from a continuous time-based function. These techniques have been implemented using Fourier transforms in transponder satellites using digital channelizers. However, no known technique has used channelizing techniques as a means for compacting data.
It would be desirable to have apparatus, methods, techniques or algorithms for compacting (reducing the size) and decompacting (recovering) time domain data with minimal distortion.